Best prediction in linear models with mixed integer/real unknowns: theory and application

In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006).     

Recursive algorithm for fast GNSS orbit fitting

Gaussian elimination is an efficient and numerically stable algorithm for estimating parameters and their precision. However, before estimating the parameters, it is often prudent to perform statistical tests to achieve the best fitting model. We use Gaussian elimination to select the best fitting model among candidate models. A succinct relationship between the weighted sum of squared residuals and the previous one is revealed by a volume formula. For quick parameter estimation and determination of weighted sum of squared residuals, a recursive elimination algorithm is proposed in the context of Gaussian elimination. In order to improve the model selection efficiency, the parameter estimation and the determination of the weighted sum of squared residuals are carried out in parallel using the proposed recursive elimination algorithm in which the improvement at each recursive stage is judged by the Bayesian information criterion. Ultimately, the computational complexity and numerical stability of the recursive elimination proposed are briefly discussed, and a GNSS orbit interpolation example is used to verify the results. It shows that the proposed recursive elimination algorithm inherits the numerical stability of the Gaussian elimination, and this algorithm can be used to examine the gain from the newly introduced parameter, dynamically assess the fitting model, and fix the optimal model efficiently. The optimal fitting model with the lowest information is very close to the real situation verified by checkpoints.     

污染误差模型下的测量数据处理理论

本文首先研究了污染误差模型的各种具体的误差表示形式,然后研究了误差服从污染误差模型时的平差准则。指出,当误差服从污染误差模型时选择均方差作为估计准则是合理的,并且符合传统的测量误差处理的观念。最后,推导了误差服从污染误差模型时,均方误差准则下的最佳估计,从而建立以均方误差准则为基础的污染误差模型下的测量数据处理理论。     

A type of biased estimators for linear models with uniformly biased data

The objective of this paper is the comparison of various types of estimators that can be used in linear models with uniformly biased data. This particular case refers to adjustment problems where the available measurements are affected by a common, unknown and uniform offset. The classic least-squares (LS) unbiased estimators for this type of models are reviewed in detail, and some additional remarks on their properties and performance are given. Furthermore, a family of biased estimators for linear models with uniformly biased data is introduced, which has the potential to provide better performance (in terms of mean squared estimation error) than the ordinary LS unbiased solutions. A number of different regularization viewpoints that can be equivalently associated with these biased estimators are presented, along with a discussion on various selection strategies that can be employed for the choice of the regularization parameter that enters into the biased estimation algorithm.     

Least-squares collocation with covariance-matching constraints

Most geostatistical methods for spatial random field (SRF) prediction using discrete data, including least-squares collocation (LSC) and the various forms of kriging, rely on the use of prior models describing the spatial correlation of the unknown field at hand over its domain. Based upon an optimal criterion of maximum local accuracy, LSC provides an unbiased field estimate that has the smallest mean squared prediction error, at every computation point, among any other linear prediction method that uses the same data. However, LSC field estimates do not reproduce the spatial variability which is implied by the adopted covariance (CV) functions of the corresponding unknown signals. This smoothing effect can be considered as a critical drawback in the sense that the spatio-statistical structure of the unknown SRF (e.g., the disturbing potential in the case of gravity field modeling) is not preserved during its optimal estimation process. If the objective for estimating a SRF from its observed functionals requires spatial variability to be represented in a pragmatic way then the results obtained through LSC may pose limitations for further inference and modeling in Earth-related physical processes, despite their local optimality in terms of minimum mean squared prediction error. The aim of this paper is to present an approach that enhances LSC-based field estimates by eliminating their inherent smoothing effect, while preserving most of their local prediction accuracy. Our methodology consists of correcting a posteriori the optimal result obtained from LSC in such a way that the new field estimate matches the spatial correlation structure implied by the signal CV function. Furthermore, an optimal criterion is imposed on the CV-matching field estimator that minimizes the loss in local prediction accuracy (in the mean squared sense) which occurs when we transform the LSC solution to fit the spatial correlation of the underlying SRF.     

Nonstationary Spatial Texture Estimation Applied to Adaptive Speckle Reduction of SAR Data

This letter proposes a new model for the second-order statistics of spatial texture in synthetic aperture radar images. The autocovariance function is locally approximated by a two-dimensional anisotropic Gaussian kernel (AGK) to characterize texture by its local orientation and anisotropy. The estimation of texture parameters at a given scale is based on the gradient structure tensor operator and does not require the explicit computation of the autocovariance. Finally, a new filter called AGK minimum mean square error (MMSE) that takes into account this spatial information is introduced and compared with the refined MMSE filter. The proposed filter has better performance in terms of texture preservation and structure enhancement     

一种稳健性准则及相应的稳健估计

研究了已有的各种稳健性度量，根据稳健性的经典定义，本文建立了一种衡量稳健性的准则，根据这一准则，研究了在测量中常用的两种误差模型下的稳健结构。结果表明，在这一个准则和随机误差模型下的最优稳健估计，是李德仁教授提出的验后方差法；在这一准则与均值移动误差模型下的最优稳健估计，是具有均方误差最小的稳健估计。     

p—范分布的近似表示

p-范分布是一个包含拉普拉斯分布、正态分析、均匀分布等常见分布的分布族。用p-范分布描述观测误差的统计特性，只需假定误差的分布为单峰、对称，因此、p-范分布似然平差可以避免事先假定误差的具体分布模式，而在平差过程中确定未知参数及误差的分布具有自适应的特点。但是p-范分布的密度函数比较复杂，不利于理论分析和实际应用。 的研究表明，p-范分布可以近似地表示为拉普拉斯分布与正态分析或正态分布均均匀分布的线性组全。p-范分布与本文给出的近似分布具有相的前四阶矩。由于拉普拉斯分布。正态分布。均匀分布的密度函数都比较简单，用近似分布代替p-范分布会使相关的问题得到简化。     

模型误差平差补偿方法比较

在高精度数据处理中,模型误差是不可忽视的。针对模型误差,简单介绍了模型误差补偿的四种方法。通过某一实例四种方法的补偿比较发现在此例中最小二乘方法补偿效果最好,并分析解释了半参数法在此例中不适合的原因。     

APPROXIMATE REPRESENTATION OF THE p-NORM DISTRIBUTION

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基于误差补偿预测树的多光谱遥感图像无损压缩方法

预测树方法是一种有效的无损多光谱图像压缩技术,将自适应线性预测方法与传统预测树方法相结合,提出了一种多光谱遥感图像的误差补偿预测树压缩方法。该方法利用多光谱图像谱间的局部统计冗余和结构冗余建立自适应预测器,对传统预测树方法产生的误差进行补偿,从而进一步减少了多光谱图像的数据量;并且利用多光谱图像的局部平稳特性对算法进行了简化。实验结果表明,该方法得到的压缩比与原始预测树方法相比有明显提高,同时算法简化后可以使计算复杂度大幅度降低。     

曲线最佳拟合的评价标准

本文探讨了曲线拟合模型选择的评价标准。主要从两方面来评价拟合质量,一方面是要使得数据拟合误差要尽量的小,另一方面要保证曲线的线形形状最佳,由此提出了新的评价准则,并通过实验验证了此评价标准的可行性。     

Biases and accuracy of, and an alternative to, discrete nonlinear filters

The biases and accuracy of the extended Kalman filter (EKF) and a second-order nonlinear filter (SONF) are discussed from the point of view of a frequentist; these are often derived by applying the relevant conditional quantities to the linear Kalman algorithm under the Bayesian framework. The EKF and the SONF are biased, although the SONF has been derived in the hope of improving first-order filters. Unfortunately the biases of the SONF may be magnified further, because the second-order terms of the relevant Bayesian conditional quantities have never been properly used to derive the SONF from the frequentist point of view. The variance–covariance matrix of the SONF given in the literature is proven to be incorrect up to the second-order approximation, and the correct one is derived. Finally, also from the point of view of a frequentist, an alternative, almost unbiased SONF is proposed, if the randomness of partials is neglected. Received: 12 July 1997 / Accepted: 5 October 1998     

基于参数统计的DEM粗差探测算法

基于参数统计的DEM粗差探测算法利用双线性内插法计算某点的高程估值,并以此进行粗差检测,方法简单易行。文中以Kriging法取代原有的内插算法,使高程估值的计算更加符合实际的分布。试验证明基于半变异函数的Kriging内插法较传统方法更为准确,也使原有算法的可靠性得到进一步的提高。     

基于偏差矫正的有偏估计及其在GPS快速定位中的应用

基于偏差矫正的一般理论提出了不适定问题的新的有偏估计。在病态条件下,Gauss-Markov模型参数的最优线性无偏估计,即LS估计是不稳健的,所得估值方差较大,严重偏离真值。因此,文中放弃了对参数估计无偏性的限制,考虑有偏估计的偏差,结合偏差矫正的正则化解法的一般理论提出了一种新的基于偏差矫正的有偏估计;结合岭估计中参数的选择方法确定了替代矩阵。最后通过GPS动态定位算例,验证了新估计的稳定性和有效性。     

干涉SAR的二维相位展开算法研究

该文讨论了干涉合成孔径雷达的二维相位展开算法。先简要给出了ＩＮＳＡＲ相位差图误差的统计特性，并说明了该误差可以近似为高斯型噪声。然后介绍了利用ＦＦＴ的最小二乘和二维相位展开算法，实验表明ＬＳ－ＦＦＴ算法对高斯白噪声具有很好的抗干扰性。     

基于ARMA模型的BDS卫星钟差异常值探测及其短期预报

星载原子钟在运行过程中会受到恶劣空间环境与设备老化等因素的影响,使得卫星钟差数据中经常存在异常值,其中AO(additive outlier)类异常值是钟差序列中常见的一类异常值.结合最大期望算法与自回归滑动平均(autoregressive moving average, ARMA)模型,提出一种AO类异常值探测算法...     

Approximate representation of thep-norm distribution

［1］Liu D J,Shi W Z,Tong X H,et al.Precision analysis and quality cont rol of GIS spatial data.Shanghai:Shanghai Publishing House of Scientific Documen ts,1999 ［2］Chen X R,Fang Z B,Li G Y,et al.Non_parameter statistics.S hanghai:Shanghai Publishing House of Science and Technology,1989 ［3］Li Q H,Tao B Z.Application of probability statistical theory in survey ing.Beijing:Beijing Publishing House of Surveying and Mapping,1982 ［4］Sun H Y.p_norm distribution theory and its application in surveyin g data processing:[Ph.D Thesis].Wuhan:Wuhan Technical University of Surveying and Mapping,1995     

误差分布的解析似合

针对根据误差值绘制直方图或用分布拟合检验法难以获得误差分布的具体类型，且方法的较为繁琐与实施不便的缺点，在分析了误差分布可用概括分布－指数分布来描述的基础上，提出了用解析法来确定误差分布的分布类型，并通过实例证明了该方法的正确性与可行性。